Step | Hyp | Ref
| Expression |
1 | | prdsbas.p |
. . . 4
β’ π = (πXsπ
) |
2 | | eqid 2733 |
. . . 4
β’
(Baseβπ) =
(Baseβπ) |
3 | | eqidd 2734 |
. . . 4
β’ (π β dom π
= dom π
) |
4 | | eqidd 2734 |
. . . 4
β’ (π β Xπ₯ β
dom π
(Baseβ(π
βπ₯)) = Xπ₯ β dom π
(Baseβ(π
βπ₯))) |
5 | | eqidd 2734 |
. . . 4
β’ (π β (π β Xπ₯ β dom π
(Baseβ(π
βπ₯)), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ (π₯ β dom π
β¦ ((πβπ₯)(+gβ(π
βπ₯))(πβπ₯)))) = (π β Xπ₯ β dom π
(Baseβ(π
βπ₯)), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ (π₯ β dom π
β¦ ((πβπ₯)(+gβ(π
βπ₯))(πβπ₯))))) |
6 | | eqidd 2734 |
. . . 4
β’ (π β (π β Xπ₯ β dom π
(Baseβ(π
βπ₯)), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ (π₯ β dom π
β¦ ((πβπ₯)(.rβ(π
βπ₯))(πβπ₯)))) = (π β Xπ₯ β dom π
(Baseβ(π
βπ₯)), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ (π₯ β dom π
β¦ ((πβπ₯)(.rβ(π
βπ₯))(πβπ₯))))) |
7 | | eqidd 2734 |
. . . 4
β’ (π β (π β (Baseβπ), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ (π₯ β dom π
β¦ (π( Β·π
β(π
βπ₯))(πβπ₯)))) = (π β (Baseβπ), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ (π₯ β dom π
β¦ (π( Β·π
β(π
βπ₯))(πβπ₯))))) |
8 | | eqidd 2734 |
. . . 4
β’ (π β (π β Xπ₯ β dom π
(Baseβ(π
βπ₯)), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ (π Ξ£g (π₯ β dom π
β¦ ((πβπ₯)(Β·πβ(π
βπ₯))(πβπ₯))))) = (π β Xπ₯ β dom π
(Baseβ(π
βπ₯)), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ (π Ξ£g (π₯ β dom π
β¦ ((πβπ₯)(Β·πβ(π
βπ₯))(πβπ₯)))))) |
9 | | eqidd 2734 |
. . . 4
β’ (π β
(βtβ(TopOpen β π
)) = (βtβ(TopOpen
β π
))) |
10 | | eqidd 2734 |
. . . 4
β’ (π β {β¨π, πβ© β£ ({π, π} β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β§ βπ₯ β dom π
(πβπ₯)(leβ(π
βπ₯))(πβπ₯))} = {β¨π, πβ© β£ ({π, π} β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β§ βπ₯ β dom π
(πβπ₯)(leβ(π
βπ₯))(πβπ₯))}) |
11 | | eqidd 2734 |
. . . 4
β’ (π β (π β Xπ₯ β dom π
(Baseβ(π
βπ₯)), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ sup((ran (π₯ β dom π
β¦ ((πβπ₯)(distβ(π
βπ₯))(πβπ₯))) βͺ {0}), β*, < ))
= (π β Xπ₯ β
dom π
(Baseβ(π
βπ₯)), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ sup((ran (π₯ β dom π
β¦ ((πβπ₯)(distβ(π
βπ₯))(πβπ₯))) βͺ {0}), β*, <
))) |
12 | | eqidd 2734 |
. . . 4
β’ (π β (π β Xπ₯ β dom π
(Baseβ(π
βπ₯)), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ Xπ₯ β dom π
((πβπ₯)(Hom β(π
βπ₯))(πβπ₯))) = (π β Xπ₯ β dom π
(Baseβ(π
βπ₯)), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ Xπ₯ β dom π
((πβπ₯)(Hom β(π
βπ₯))(πβπ₯)))) |
13 | | eqidd 2734 |
. . . 4
β’ (π β (π β (Xπ₯ β dom π
(Baseβ(π
βπ₯)) Γ Xπ₯ β dom π
(Baseβ(π
βπ₯))), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ (π β ((2nd βπ)(π β Xπ₯ β dom π
(Baseβ(π
βπ₯)), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ Xπ₯ β dom π
((πβπ₯)(Hom β(π
βπ₯))(πβπ₯)))π), π β ((π β Xπ₯ β dom π
(Baseβ(π
βπ₯)), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ Xπ₯ β dom π
((πβπ₯)(Hom β(π
βπ₯))(πβπ₯)))βπ) β¦ (π₯ β dom π
β¦ ((πβπ₯)(β¨((1st βπ)βπ₯), ((2nd βπ)βπ₯)β©(compβ(π
βπ₯))(πβπ₯))(πβπ₯))))) = (π β (Xπ₯ β dom π
(Baseβ(π
βπ₯)) Γ Xπ₯ β dom π
(Baseβ(π
βπ₯))), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ (π β ((2nd βπ)(π β Xπ₯ β dom π
(Baseβ(π
βπ₯)), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ Xπ₯ β dom π
((πβπ₯)(Hom β(π
βπ₯))(πβπ₯)))π), π β ((π β Xπ₯ β dom π
(Baseβ(π
βπ₯)), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ Xπ₯ β dom π
((πβπ₯)(Hom β(π
βπ₯))(πβπ₯)))βπ) β¦ (π₯ β dom π
β¦ ((πβπ₯)(β¨((1st βπ)βπ₯), ((2nd βπ)βπ₯)β©(compβ(π
βπ₯))(πβπ₯))(πβπ₯)))))) |
14 | | prdsbas.s |
. . . 4
β’ (π β π β π) |
15 | | prdsbas.r |
. . . 4
β’ (π β π
β π) |
16 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15 | prdsval 17342 |
. . 3
β’ (π β π = (({β¨(Baseβndx), Xπ₯ β
dom π
(Baseβ(π
βπ₯))β©, β¨(+gβndx),
(π β Xπ₯ β
dom π
(Baseβ(π
βπ₯)), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ (π₯ β dom π
β¦ ((πβπ₯)(+gβ(π
βπ₯))(πβπ₯))))β©, β¨(.rβndx),
(π β Xπ₯ β
dom π
(Baseβ(π
βπ₯)), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ (π₯ β dom π
β¦ ((πβπ₯)(.rβ(π
βπ₯))(πβπ₯))))β©} βͺ {β¨(Scalarβndx),
πβ©, β¨(
Β·π βndx), (π β (Baseβπ), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ (π₯ β dom π
β¦ (π( Β·π
β(π
βπ₯))(πβπ₯))))β©,
β¨(Β·πβndx), (π β Xπ₯ β dom π
(Baseβ(π
βπ₯)), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ (π Ξ£g (π₯ β dom π
β¦ ((πβπ₯)(Β·πβ(π
βπ₯))(πβπ₯)))))β©}) βͺ ({β¨(TopSetβndx),
(βtβ(TopOpen β π
))β©, β¨(leβndx), {β¨π, πβ© β£ ({π, π} β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β§ βπ₯ β dom π
(πβπ₯)(leβ(π
βπ₯))(πβπ₯))}β©, β¨(distβndx), (π β Xπ₯ β dom π
(Baseβ(π
βπ₯)), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ sup((ran (π₯ β dom π
β¦ ((πβπ₯)(distβ(π
βπ₯))(πβπ₯))) βͺ {0}), β*, <
))β©} βͺ {β¨(Hom βndx), (π β Xπ₯ β dom π
(Baseβ(π
βπ₯)), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ Xπ₯ β dom π
((πβπ₯)(Hom β(π
βπ₯))(πβπ₯)))β©, β¨(compβndx), (π β (Xπ₯ β dom
π
(Baseβ(π
βπ₯)) Γ Xπ₯ β dom π
(Baseβ(π
βπ₯))), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ (π β ((2nd βπ)(π β Xπ₯ β dom π
(Baseβ(π
βπ₯)), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ Xπ₯ β dom π
((πβπ₯)(Hom β(π
βπ₯))(πβπ₯)))π), π β ((π β Xπ₯ β dom π
(Baseβ(π
βπ₯)), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ Xπ₯ β dom π
((πβπ₯)(Hom β(π
βπ₯))(πβπ₯)))βπ) β¦ (π₯ β dom π
β¦ ((πβπ₯)(β¨((1st βπ)βπ₯), ((2nd βπ)βπ₯)β©(compβ(π
βπ₯))(πβπ₯))(πβπ₯)))))β©}))) |
17 | | eqid 2733 |
. . 3
β’
(Scalarβπ) =
(Scalarβπ) |
18 | | scaid 17201 |
. . 3
β’ Scalar =
Slot (Scalarβndx) |
19 | | snsstp1 4777 |
. . . . 5
β’
{β¨(Scalarβndx), πβ©} β {β¨(Scalarβndx),
πβ©, β¨(
Β·π βndx), (π β (Baseβπ), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ (π₯ β dom π
β¦ (π( Β·π
β(π
βπ₯))(πβπ₯))))β©,
β¨(Β·πβndx), (π β Xπ₯ β dom π
(Baseβ(π
βπ₯)), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ (π Ξ£g (π₯ β dom π
β¦ ((πβπ₯)(Β·πβ(π
βπ₯))(πβπ₯)))))β©} |
20 | | ssun2 4134 |
. . . . 5
β’
{β¨(Scalarβndx), πβ©, β¨(
Β·π βndx), (π β (Baseβπ), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ (π₯ β dom π
β¦ (π( Β·π
β(π
βπ₯))(πβπ₯))))β©,
β¨(Β·πβndx), (π β Xπ₯ β dom π
(Baseβ(π
βπ₯)), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ (π Ξ£g (π₯ β dom π
β¦ ((πβπ₯)(Β·πβ(π
βπ₯))(πβπ₯)))))β©} β ({β¨(Baseβndx),
Xπ₯ β
dom π
(Baseβ(π
βπ₯))β©, β¨(+gβndx), (π β Xπ₯ β dom π
(Baseβ(π
βπ₯)), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ (π₯ β dom π
β¦ ((πβπ₯)(+gβ(π
βπ₯))(πβπ₯))))β©, β¨(.rβndx),
(π β Xπ₯ β dom
π
(Baseβ(π
βπ₯)), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ (π₯ β dom π
β¦ ((πβπ₯)(.rβ(π
βπ₯))(πβπ₯))))β©} βͺ {β¨(Scalarβndx), πβ©, β¨(
Β·π βndx), (π β (Baseβπ), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ (π₯ β dom π
β¦ (π( Β·π
β(π
βπ₯))(πβπ₯))))β©,
β¨(Β·πβndx), (π β Xπ₯ β dom π
(Baseβ(π
βπ₯)), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ (π Ξ£g (π₯ β dom π
β¦ ((πβπ₯)(Β·πβ(π
βπ₯))(πβπ₯)))))β©}) |
21 | 19, 20 | sstri 3954 |
. . . 4
β’
{β¨(Scalarβndx), πβ©} β ({β¨(Baseβndx),
Xπ₯
β dom π
(Baseβ(π
βπ₯))β©, β¨(+gβndx),
(π β Xπ₯ β
dom π
(Baseβ(π
βπ₯)), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ (π₯ β dom π
β¦ ((πβπ₯)(+gβ(π
βπ₯))(πβπ₯))))β©, β¨(.rβndx),
(π β Xπ₯ β
dom π
(Baseβ(π
βπ₯)), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ (π₯ β dom π
β¦ ((πβπ₯)(.rβ(π
βπ₯))(πβπ₯))))β©} βͺ {β¨(Scalarβndx),
πβ©, β¨(
Β·π βndx), (π β (Baseβπ), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ (π₯ β dom π
β¦ (π( Β·π
β(π
βπ₯))(πβπ₯))))β©,
β¨(Β·πβndx), (π β Xπ₯ β dom π
(Baseβ(π
βπ₯)), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ (π Ξ£g (π₯ β dom π
β¦ ((πβπ₯)(Β·πβ(π
βπ₯))(πβπ₯)))))β©}) |
22 | | ssun1 4133 |
. . . 4
β’
({β¨(Baseβndx), Xπ₯ β dom π
(Baseβ(π
βπ₯))β©, β¨(+gβndx),
(π β Xπ₯ β
dom π
(Baseβ(π
βπ₯)), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ (π₯ β dom π
β¦ ((πβπ₯)(+gβ(π
βπ₯))(πβπ₯))))β©, β¨(.rβndx),
(π β Xπ₯ β
dom π
(Baseβ(π
βπ₯)), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ (π₯ β dom π
β¦ ((πβπ₯)(.rβ(π
βπ₯))(πβπ₯))))β©} βͺ {β¨(Scalarβndx),
πβ©, β¨(
Β·π βndx), (π β (Baseβπ), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ (π₯ β dom π
β¦ (π( Β·π
β(π
βπ₯))(πβπ₯))))β©,
β¨(Β·πβndx), (π β Xπ₯ β dom π
(Baseβ(π
βπ₯)), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ (π Ξ£g (π₯ β dom π
β¦ ((πβπ₯)(Β·πβ(π
βπ₯))(πβπ₯)))))β©}) β (({β¨(Baseβndx),
Xπ₯ β
dom π
(Baseβ(π
βπ₯))β©, β¨(+gβndx), (π β Xπ₯ β dom π
(Baseβ(π
βπ₯)), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ (π₯ β dom π
β¦ ((πβπ₯)(+gβ(π
βπ₯))(πβπ₯))))β©, β¨(.rβndx),
(π β Xπ₯ β dom
π
(Baseβ(π
βπ₯)), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ (π₯ β dom π
β¦ ((πβπ₯)(.rβ(π
βπ₯))(πβπ₯))))β©} βͺ {β¨(Scalarβndx), πβ©, β¨(
Β·π βndx), (π β (Baseβπ), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ (π₯ β dom π
β¦ (π( Β·π
β(π
βπ₯))(πβπ₯))))β©,
β¨(Β·πβndx), (π β Xπ₯ β dom π
(Baseβ(π
βπ₯)), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ (π Ξ£g (π₯ β dom π
β¦ ((πβπ₯)(Β·πβ(π
βπ₯))(πβπ₯)))))β©}) βͺ ({β¨(TopSetβndx),
(βtβ(TopOpen β π
))β©, β¨(leβndx), {β¨π, πβ© β£ ({π, π} β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β§ βπ₯ β dom π
(πβπ₯)(leβ(π
βπ₯))(πβπ₯))}β©, β¨(distβndx), (π β Xπ₯ β dom π
(Baseβ(π
βπ₯)), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ sup((ran (π₯ β dom π
β¦ ((πβπ₯)(distβ(π
βπ₯))(πβπ₯))) βͺ {0}), β*, <
))β©} βͺ {β¨(Hom βndx), (π β Xπ₯ β dom π
(Baseβ(π
βπ₯)), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ Xπ₯ β dom π
((πβπ₯)(Hom β(π
βπ₯))(πβπ₯)))β©, β¨(compβndx), (π β (Xπ₯ β dom
π
(Baseβ(π
βπ₯)) Γ Xπ₯ β dom π
(Baseβ(π
βπ₯))), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ (π β ((2nd βπ)(π β Xπ₯ β dom π
(Baseβ(π
βπ₯)), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ Xπ₯ β dom π
((πβπ₯)(Hom β(π
βπ₯))(πβπ₯)))π), π β ((π β Xπ₯ β dom π
(Baseβ(π
βπ₯)), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ Xπ₯ β dom π
((πβπ₯)(Hom β(π
βπ₯))(πβπ₯)))βπ) β¦ (π₯ β dom π
β¦ ((πβπ₯)(β¨((1st βπ)βπ₯), ((2nd βπ)βπ₯)β©(compβ(π
βπ₯))(πβπ₯))(πβπ₯)))))β©})) |
23 | 21, 22 | sstri 3954 |
. . 3
β’
{β¨(Scalarβndx), πβ©} β (({β¨(Baseβndx),
Xπ₯
β dom π
(Baseβ(π
βπ₯))β©, β¨(+gβndx),
(π β Xπ₯ β
dom π
(Baseβ(π
βπ₯)), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ (π₯ β dom π
β¦ ((πβπ₯)(+gβ(π
βπ₯))(πβπ₯))))β©, β¨(.rβndx),
(π β Xπ₯ β
dom π
(Baseβ(π
βπ₯)), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ (π₯ β dom π
β¦ ((πβπ₯)(.rβ(π
βπ₯))(πβπ₯))))β©} βͺ {β¨(Scalarβndx),
πβ©, β¨(
Β·π βndx), (π β (Baseβπ), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ (π₯ β dom π
β¦ (π( Β·π
β(π
βπ₯))(πβπ₯))))β©,
β¨(Β·πβndx), (π β Xπ₯ β dom π
(Baseβ(π
βπ₯)), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ (π Ξ£g (π₯ β dom π
β¦ ((πβπ₯)(Β·πβ(π
βπ₯))(πβπ₯)))))β©}) βͺ ({β¨(TopSetβndx),
(βtβ(TopOpen β π
))β©, β¨(leβndx), {β¨π, πβ© β£ ({π, π} β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β§ βπ₯ β dom π
(πβπ₯)(leβ(π
βπ₯))(πβπ₯))}β©, β¨(distβndx), (π β Xπ₯ β dom π
(Baseβ(π
βπ₯)), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ sup((ran (π₯ β dom π
β¦ ((πβπ₯)(distβ(π
βπ₯))(πβπ₯))) βͺ {0}), β*, <
))β©} βͺ {β¨(Hom βndx), (π β Xπ₯ β dom π
(Baseβ(π
βπ₯)), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ Xπ₯ β dom π
((πβπ₯)(Hom β(π
βπ₯))(πβπ₯)))β©, β¨(compβndx), (π β (Xπ₯ β dom
π
(Baseβ(π
βπ₯)) Γ Xπ₯ β dom π
(Baseβ(π
βπ₯))), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ (π β ((2nd βπ)(π β Xπ₯ β dom π
(Baseβ(π
βπ₯)), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ Xπ₯ β dom π
((πβπ₯)(Hom β(π
βπ₯))(πβπ₯)))π), π β ((π β Xπ₯ β dom π
(Baseβ(π
βπ₯)), π β Xπ₯ β dom π
(Baseβ(π
βπ₯)) β¦ Xπ₯ β dom π
((πβπ₯)(Hom β(π
βπ₯))(πβπ₯)))βπ) β¦ (π₯ β dom π
β¦ ((πβπ₯)(β¨((1st βπ)βπ₯), ((2nd βπ)βπ₯)β©(compβ(π
βπ₯))(πβπ₯))(πβπ₯)))))β©})) |
24 | 16, 17, 18, 14, 23 | prdsbaslem 17340 |
. 2
β’ (π β (Scalarβπ) = π) |
25 | 24 | eqcomd 2739 |
1
β’ (π β π = (Scalarβπ)) |